Aiming for Maximum Tracking Accuracy in Repetitive Control Systems

This paper shows that design of a robustly stable repetitive control system is equivalent to that of a feedback control system for an uncertain linear time-invariant system satisfying the well-known robust performance condition. Once a feedback controller is designed to satisfy the robust performance condition, the feedback controller and the repetitive controller using the performance weighting function robustly stabilizes the repetitive control system. It is also shown that we can obtain a steady-state tracking error described in a simple form without time-delay element if the robust stability condition is satisfied for the repetitive control system. Moreover, using this result, a sufficient condition is provided, which ensures that the least upper bound of the steady-state tracking error generated by the repetitive control system is less than or equal to the least upper bound of the steady-state tracking error only by the feedback system

The modified repetitive control system is a type of servomechanism for periodic reference inputs. That is, the modified repetitive control system follows the periodic reference input with a small steady state error, even if a periodic disturbance or uncertainty exists in the plant. Using previously proposed modified repetitive controllers, even if the plant does not include time-delay, transfer functions from the periodic reference input to the output and from the disturbance to the output have infmite numbers of poles. To specify the input-output and the disturbance attenuation characteristic easily, Yamada et al. proposed the concept of a simple repetitive control system such that the controller works as a modified repetitive controller and transfer functions from the periodic reference input to the output and from the disturbance to the output have fmite numbers of poles. In addition, Yamada et al. clarified the parameterization of all stabilizing simple repetitive controllers. Recently, the p...

In view of the surplus torque and complexity of controlled plant in passive electrical load system, a novel approach based on single neuron PID adaptive control and repetitive control for repetitive periodic load control system is proposed. Radial basis function (RBF) neural network is used to identify the system on-line for the single neuron PID controller to adjust its weights and PID parameters by self- learning and self-adapting based on the desired output. The dynamic state performance can be improved by the single neuron adaptive PID control and the steady state performance is also improved by modified repetitive control. Computer simulation results show that the force/position hybrid control system can effectively reduce the surplus torque and improve the loading precision, and also it has fine dynamic and steady state performance and good robustness. The reliability of whole system is further improved.

Effects of Non-Periodic Disturbances on Repetitive Control Systems

Repetitive control is one control algorithm based on the Internal Model Principle (Francis & Wonham, 1976) and has been widely implemented in various applications. A repetitive control based system has been shown to work well for tracking periodic reference commands or for rejecting periodic disturbances. Although the idea has been verified as early as 1981 (Inoue et al., 1981), a rigorous analysis and synthesis of repetitive controllers for continuous-time systems was not proposed until 1989, by Hara et al. (Hara et al., 1988). Tomizuka et al. (Tomizuka et al., 1989) addressed the analysis and synthesis of discrete-time repetitive controller, considering the fact that digital implementation of a repetitive controller is simpler and more straightforward. Since then, repetitive control has gained popularity in applications where periodic disturbances rejection or repetitive tracking are required, see (Wang et al., 2009; Cuiyan et al., 2004) and the references therein. These include controls of disk drive servo (Tomizuka et al., 1989; Guo, 1997; Moon et al., 1998), hydraulic closed-loop servo for material testing (Srinivasan & Shaw, 1993), vibration suppression (Hillerstrom, 1996), rejection of load disturbances in steel casting process (Manayathara et al., 1996), servo control for a positioning table (Yamada et al., 1999), X-Y table (Tung et al., 1993), noncircular turning process (Alter & Tsao, 1994), motor speed ripple reduction (Godler et al., 1995; Rodriguez et al., 2000), and eccentricity compensation (Garimella & Srinivasan, 1996). In literatures, repetitive controllers are synthesized and operate in time domain, which is in accordance with the fact that models or differential equations of physical systems are mostly derived using time as the independent variable. One of the key steps for designing a repetitive controller is to determine the period, or equivalently, the number of delay taps (q−1, q is the one step advance operator). This can usually be done by analyzing the periodic tracking or disturbance signal using techniques such as fast Fourier transform (FFT). To ensure effectiveness of the design, an underlying assumption is that the frequency constitutions of the periodic tracking or disturbance signal do not vary with respect to time, which corresponds to a stationary or time-invariant frequency spectrum of the signal. This assumption can be satisfied when the design objective is to track a prespecified periodic trajectory. However, it might be violated for disturbance rejection problems where the frequency constitutions of the disturbance are time-varying. For a motion system with rotary components such as gear-train, the disturbances due to gear

We give a parameterization of all causal stabilizing repetitive learning controllers for single-input and single-output linear minimum phase systems. The past studies on the parameterization of a repetitive learning control system were to obtain the parameterization of all internally stabilizing controllers for the control system such that the plant is connected 1/(1-e/sup -sT/) in series. Since 1/(1-e/sup -sT/) has no zeros in the closed right half plane, also the serial connections of 1/(1-e/sup -sT/) and a minimum phase system have no zeros in the closed right half plane. That is, the parameterization of all stabilizing controllers for a repetitive learning control system may be obtained by using the parameterization for a minimum phase system in Glaria and Goodwin (1994). Using the above mentioned idea, at first, the parameterization of all stabilizing repetitive learning controllers for the biproper minimum phase system is carried out. Next, since the exact repetitive learning controllers to stabilize the strictly stable proper system do not exist, the parameterization of all modified repetitive learning controllers using a low-pass filter in the repetitive learning controller is given.

The paper presents a limited angle torque motor used in optical scanning system. A method called repetitive control, in which the controlled variables follow periodic reference commands, is proposed to design the limited angle torque motor tracking control system. The compensator in the repetitive control system is synthesized by H/sub /spl infin// optimal control design theory. Experimental results show that the tracking performance of the repetitive controller is superior to that of the conventional controller.

A new design method of the repetitive control system for linear systems with time-varying norm-bounded uncertainties is presented, which can design a low-pass filter of the repetitive controller and a stabilizing controller simultaneously in the repetitive control system. Using a Lyapunov functional for the repetitive control system, sufficient conditions are derived in terms of a linear matrix inequality (LMI), which ensure robust stability of the repetitive control system and achieve the prescribed H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> performance. Based on derived conditions, the problem of designing a stabilizing controller and finding the cut-off frequency of a low-pass filter is reformulated as an optimization problem with an LMI constraint of the free parameter. Then, an iterative algorithm is presented to calculate the largest cut-off frequency of a low-pass filter and design the corresponding state-feedback controller simultaneously. A numerical example is presented to illustrate the validity of the proposed method.

The problem of designing output error controllers to track periodic reference signals for a class of uncertain linear time-delay systems subject to periodic disturbances is addressed. A repetitive controller, which is a type of output error controller, is developed for these systems and a sufficient condition for stability of the closed-loop system is derived by applying the Lyapunov–Krasovskii functional and linear matrix inequalities (LMIs). In order to relax the stability condition, the theory of the modified repetitive control (RC) system is extended to the systems represented by irrational transfer functions. Based on the extended theory of the modified RC system, a modified repetitive controller, another type of output error controller, is developed for these systems and sufficient conditions for stability of the closed-loop system are derived in terms of a frequency-domain criterion and LMI. Numerical simulations are presented to demonstrate the effectiveness of the proposed controllers.

It is well known that the speed control system, even though which is constructed with an excellent control method, has an unexpected steady speed error when it is suffered by a periodical load disturbance. As the method of eliminating the speed fluctuation caused by such kind of disturbance, a repetitive control is the most effective. But the repetitive control method is only adoptive for a constant speed control system.In this paper, a novel control system designing method which is effective to both the constant speed control system and variable speed driving system is shown. The fact that a periodical disturbance depends on time function and also rotational angle is utilized to design the control system. From the result, the numerical modeling of the controlled object is derived in the rotational angle function domain. For the obtained numerical model, an extended state variable equation based on the original system is presented by using the control technique named “Error System Technique”. And then the optimal feedback gains are decided by solving the optimal regulator problem for the extended state equation. Therefore, a stability of the system is ensured.The proposed control method is applied to DC motor variable speed control system and the experimental setup is constructed. Finally, the effectiveness of the proposed method is confirmed by the result of simulations and experiments.

This paper describes the design and implementation of a repetitive controller (RC) for tracking periodic trajectories in aerial robots, such as unmanned aerial vehicles (UAVs). The RC is ideally suited for situations where the robot is required to execute a defined flight pattern, over and over again in a periodic fashion. Applications include precision agriculture, and search and rescue, where the desired trajectory is often periodic in time and known a priori. Compared to iterative learning control (ILC), RC is a feedback-based approach which can be plugged into an existing feedback control loop to improve the tracking of period trajectories. Herein, a plug-in type RC is applied to a micro aerial robot, and a systematic approach is presented for designing the control system. The plug-in nature of the design also allows users to enable/disable the controller on the fly. The proposed RC is also compared to existing approaches, including no-reset ILC, which have recently been applied to control aerial robots. The effectiveness of the RC is demonstrated on a custom-designed experimental micro aerial robot. Tracking of circle, square, and figure-eight trajectories are shown in both simulations and experiments. In these cases, the RC reduces the average tracking error by over 58% compared to a well-tuned PID controller, and simulations that compare to no-reset ILC show that the performance of RC is less sensitive to changes in the reference trajectory.

A repetitive control method is presented that is implemented in real-time for periodic wind disturbance rejection for linear systems with multiple inputs and multiple outputs and with both repetitive and non-repetitive disturbance components. The novel repetitive controller can reject the periodic wind disturbances for fixed-speed wind turbines and variable-speed wind turbines operating above-rated and we will demonstrate this on an experimental “smart” rotor test section. The “smart” rotor is a rotor where the blades are equipped with a number of control devices that locally change the lift profile on the blade, combined with appropriate sensors and controllers. The rotational speed of wind turbines operating above-rated will vary around a defined reference speed, therefore methods are given to robustify the repetitive controllers for a mismatch in the period. The design of the repetitive controller is formulated as a lifted linear stochastic output-feedback problem on which the mature techniques of discrete linear control may be applied. For real-time implementation, the computational complexity can be reduced by exploiting the structure in the lifted state-space matrices. With relatively slow changing periodic disturbances it is shown that this repetitive control method can significantly reduce the structural vibrations of the “smart” rotor test section. The cost of additional wear and tear of the “smart” actuators are kept small, because a smooth control action is generated as the controller mainly focuses on the reduction of periodic disturbances.

A repetitive control theory is proposed to solve the synchronization problem between the wafer stage and reticle stage. A macro-micro control method is used based on a macro-micro control structure in which a linear motor is combined with a voice coil motor. A synchronization controller of the reticle stage is added base on the conventional PID control system. The repetitive controller is designed based on the repeated movement of the reticle stage and the wafer stage during the scan and exposure period, and the effects of synchronization control system can be improved because of the repetitive control can effectively track and inhibit the periodicity excitation signal. The repetitive control system effectively reduces the synchronization error during the scan and exposure period，in the meanwhile keep the tracking accuracy and dynamic characters. Simulation results show that the synchronization error can be reduced effectively.

The structured singular value method is applied to continuous-time SISO repetitive control systems in order to determine their stability and performance robustness in the presence of structured parametric modeling error in the plant. Performance measures for repetitive control systems are introduced and the robust performance analysis using structured singular values is modified, taking the alternating magnitude behaviour of the sensitivity function of a repetitive control system into account. The analysis procedure is simplified considerably in the case of large time delay by noting that the upper and lower envelopes of the structured singular value plots are obtained from corresponding finite dimensional systems without any time delay. The analysis procedures developed are applied to an example on repetitive control of an electrohydraulic material testing machine, available in the literature, to demonstrate their usefulness.

Implementation of Plug-in Type Repetitive Controller for Position-based Periodic Control Systems